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In the Scriptures, "church government" is simple. Each church had a group of elders who shepherded the church and were called overseers (Gr. episkopos, usually translated, ecclesiastically, as bishop).
If we discuss very early Christian history, when the churches were small, then the elder/overseer role stays easy. Western churches, such as Rome and Philippi maintained the multiple overseer pattern we see in Paul and Peter's churches.
It appears to me that John's churches (those in Asia Minor, of which the 7 in Rev. 2-3 are examples) had a group of elders of which one and one only was the overseer (explanation here).
By the mid-second century, all churches had adopted that pattern.
It is then that church government begins to be complicated. The churches in major cities may have been quite big by then, and if they weren't by AD 150, they certainly were by AD 200.
How did church government expand to accomodate that growth? I'm not sure of all the details, but the general picture is clear.
In a big city, in the third century, there would have been multiple overseers (bishops). I'm pretty sure early in the third century, there would be one main overseer over the city, and minor overseers in the smaller towns who in some way reported to the city's overseer, who was now called a "metropolitan."
Underneath these overseers were the elders.
It seems likely, but I don't know for certain, that there were smaller congregations in the city that were led by one of the elders or maybe more than one. All the elders in the city would consider the metropolitan as their overseer/bishop.
It also seems likely that surrounding towns had multiple elders plus their overseer, who was submitted to the metropolitan.
By the late 3rd century, and confirmed at the Council of Nicea in the early 4th century, three metropolitans were given the status of "patriarch." They weren't called patriarchs at the council, but the position apart from the name was officially recognized and approbated at that time. These were metropolitans that led the synods of bishops of entire countries: Alexandria over Egypt and Libya; Rome over Italy, Carthage, and Spain (and probably the Germanic kingdoms); and Antioch with "similar" authority (Canon 6).
The fall of the western Roman empire left the Roman patriarch as the only patriarch in the west. It didn't take long for him to gain authority over all the churches and the spiritual subservience of the "Christian" kings of the barbarians. At that point, the Roman patriarch's enamor with his own authority inevitably led to the doctrine of papal primacy.
One of the things I've been asked is whether smaller congregations in one city would have been led by multiple elders, or did each congregation have one elder that made up the plurality of elders in that city.
I do not know the answer to that question. I have never seen it addressed. So, if you run across an authoritative answer, please use the "contact me" button in the navbar and let me know. I'm speaking of the situation during the third century.
This video is somewhat related to this subject, and it's one of my favorites that I've done. It's long, but it has a lot of valuable information. Even listening to part of it would be helpful.
If the video doesn't play for you on this site, just click on it, and it will send you to Youtube.